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It is well known that the density-gradient expansion of the Hartree-Pock exchange energy for the bare-Coulomb interaction contains divergent terms of order e where e is the electronic charge. We argue that the exchange energy evaluated with Kohn-Sham orbitals (i.e. , those derived from a local effective potential) is purely of order e and therefore its gradient expansion is well defined. This density-gradient expansion, with the a priori coefficient of Sham, is shown to converge by comparison with numerically refined values for the exact exchange energy of a metal surface in the linear-potential model. As the electron density profile becomes more slowly varying, the relative error of the zerothorder (local-density) term tends to zero. We present here the first demonstration that, in addition, the absolute error is increasingly canceled by the second-order {gradient) term.Like the gradient expansion for the kinetic energy but unlike the one for correlation, the gradient expansion for exchange gives useful results even for "physical" surface profiles.One-and many-electron atoms are also discussed. It is observed that, as the atomic number increases, the relative errors of the local-density and gradient-expansion approximations decrease in magnitude, but the gradient term corrects only a small fraction of the error of the local-density approximation. This is a consequence of the fact that the convergence condition~Vn~/2kFn & l is increasingly satisfied as the atomic number increases but the second convergence condition ) V n~/2kF~Vn~&& 1 is not so well satisfied.

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For a system of N electrons in an external scalar potential v(r) and external vector potential A(r), we prove that the wave function ψ is a functional of the gauge invariant ground state density ρ(r) and ground state physical current density j(r), and a gauge function α(R) (withIt is the presence of the gauge function α(R) that ensures the wave function functional is gauge variant. We prove this via a unitary transformation and by a proof of the bijectivity between the potentials {v(r), A(r)} and the ground state properties {ρ(r), j(r)}. Thus, the natural basic variables for the system are the gauge invariant ρ(r) and j(r). Because each choice of gauge function corresponds to the same physical system, the choice of α(R) = 0 is equally valid. As such, we construct a {ρ(r), j(r)} functional theory with the corresponding Euler equations for the density ρ(r) and physical current density j(r), together with the constraints of charge conservation and continuity of the current. With the assumption of existence of a system of noninteracting fermions with the same ρ(r) and j(r) as that of the electrons, we provide the equations describing this model system, the definitions being within the framework of Kohn-Sham theory in terms of energy functionals of {ρ(r), j(r)} and their functional derivatives. A special case of the {ρ(r), j(r)} functional theory is the magnetic-field density-functional theory of Grayce and Harris. We discuss and contrast our work with the paramagnetic current-and density-functional theory of Vignale and Rasolt in which the variables are the gauge invariant ground state density ρ(r), and vorticity ν(r) = ∇ × (j p (r)/ρ(r)), where j p (r) is the paramagnetic current density.

We explain by quantal density functional theory the physics of mapping from any bound nondegenerate excited state of Schrödinger theory to an S system of noninteracting fermions with equivalent density and energy. The S system may be in a ground or excited state. In either case, the highest occupied eigenvalue is the negative of the ionization potential. We demonstrate this physics with examples. The theory further provides a new framework for calculations of atomic excited states including multiplet structure.